ec9a73fd - mathematics_in

Authored by: Jeremy Avigad
Authored at: 2024-08-19 10:49:00 -0400
Committed by: Jeremy Avigad
Committed at: 2024-08-19 10:49:00 -0400
Object name: ec9a73fdfb8af2708593b307ebb3d1bd076893cc
Parent: df662034

Update Mon Aug 19 10:49:00 AM EDT 2024

Changes

44 changed files (+421/-355)

MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean
- @@ -69,10 +69,10 @@ example {m n : ℕ} (coprime_mn : m.Coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by
  example {m n p : ℕ} (coprime_mn : m.Coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by sorry - #check Nat.factors - #check Nat.prime_of_mem_factors - #check Nat.prod_factors - #check Nat.factors_unique + #check Nat.primeFactorsList + #check Nat.prime_of_mem_primeFactorsList + #check Nat.prod_primeFactorsList + #check Nat.primeFactorsList_unique theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : (m * n).factorization p = m.factorization p + n.factorization p := by
MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean
- @@ -85,8 +85,8 @@ example (f : ℕ → ℕ) (n : ℕ) : ∏ x in range n.succ, f x = (∏ x in ran
  example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := by induction' n with n ih - · rw [fac, prod_range_zero] - rw [fac, ih, prod_range_succ, mul_comm] + · simp [fac, prod_range_zero] + simp [fac, ih, prod_range_succ, mul_comm] example (a b c d e f : ℕ) : a * (b * c * f * (d * e)) = d * (a * f * e) * (c * b) := by simp [mul_assoc, mul_comm, mul_left_comm]
- @@ -102,7 +102,7 @@ theorem sum_sqr (n : ℕ) : ∑ i in range (n + 1), i ^ 2 = n * (n + 1) * (2 * n
  sorry end - inductive MyNat + inductive MyNat where | zero : MyNat | succ : MyNat → MyNat
MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean
- @@ -9,7 +9,7 @@ theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by
  cases m; contradiction case succ m => cases m; contradiction - repeat' apply Nat.succ_le_succ + repeat apply Nat.succ_le_succ apply zero_le example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by
- @@ -118,7 +118,7 @@ example (s : Finset ℕ) (x : ℕ) : x ∈ s.filter Nat.Prime ↔ x ∈ s ∧ x.
  theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by intro s by_contra h - push_neg at h + push_neg at h set s' := s.filter Nat.Prime with s'_def have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by intro n
- @@ -178,7 +178,7 @@ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) :
  · use n induction' n using Nat.strong_induction_on with n ih rw [Nat.prime_def_lt] at np - push_neg at np + push_neg at np rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ have mge2 : 2 ≤ m := by apply two_le _ mne1
- @@ -201,7 +201,7 @@ example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s
  theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by by_contra h - push_neg at h + push_neg at h rcases h with ⟨n, hn⟩ have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by apply ex_finset_of_bounded
MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean
- @@ -33,7 +33,7 @@ theorem sum_sqr (n : ℕ) : ∑ i in range (n + 1), i ^ 2 = n * (n + 1) * (2 * n
  end - inductive MyNat + inductive MyNat where | zero : MyNat | succ : MyNat → MyNat
MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
- @@ -9,7 +9,7 @@ theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by
  cases m; contradiction case succ m => cases m; contradiction - repeat' apply Nat.succ_le_succ + repeat apply Nat.succ_le_succ apply zero_le theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by
- @@ -102,7 +102,7 @@ theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) :
  theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by intro s by_contra h - push_neg at h + push_neg at h set s' := s.filter Nat.Prime with s'_def have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by intro n
- @@ -168,7 +168,7 @@ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) :
  · use n induction' n using Nat.strong_induction_on with n ih rw [Nat.prime_def_lt] at np - push_neg at np + push_neg at np rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ have mge2 : 2 ≤ m := by apply two_le _ mne1
- @@ -194,7 +194,7 @@ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) :
  theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by by_contra h - push_neg at h + push_neg at h rcases h with ⟨n, hn⟩ have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by apply ex_finset_of_bounded
MIL/C08_Groups_and_Rings/S01_Groups.lean
- @@ -145,19 +145,19 @@ example {G : Type*} [Group G] (G' : Subgroup G) : Nat.card G' ∣ Nat.card G :=
  open Subgroup - example {G : Type*} [Group G] [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime] + example {G : Type*} [Group G] [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime] (hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n := Sylow.exists_subgroup_card_pow_prime p hdvd - lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} [Fintype H] : + lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} : H = ⊥ ↔ Nat.card H = 1 := by suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by - simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists, -Nat.card_eq_fintype_card] + simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists] sorry #check card_dvd_of_le - lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) [Fintype H] [Fintype K] + lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) (h : (Nat.card H).Coprime (Nat.card K)) : H ⊓ K = ⊥ := by sorry open Equiv
- @@ -272,7 +272,7 @@ open MonoidHom
  #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right - lemma aux_card_eq [Fintype G] (h' : Nat.card G = Nat.card H * Nat.card K) : + lemma aux_card_eq [Finite G] (h' : Nat.card G = Nat.card H * Nat.card K) : Nat.card (G ⧸ H) = Nat.card K := by sorry variable [H.Normal] [K.Normal] [Fintype G] (h : Disjoint H K)
MIL/C08_Groups_and_Rings/S02_Rings.lean
- @@ -1,7 +1,7 @@
  import Mathlib.RingTheory.Ideal.QuotientOperations import Mathlib.RingTheory.Localization.Basic import Mathlib.RingTheory.DedekindDomain.Ideal - import Mathlib.Analysis.Complex.Polynomial + import Mathlib.Analysis.Complex.Polynomial.Basic import Mathlib.Data.ZMod.Quotient import MIL.Common
MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean
- @@ -71,10 +71,10 @@ open scoped Classical
  open Subgroup - lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} [Fintype H] : + lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} : H = ⊥ ↔ Nat.card H = 1 := by suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by - simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists, -Nat.card_eq_fintype_card] + simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists] constructor · intro h use 1, H.one_mem
- @@ -82,7 +82,7 @@ lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} [Fintype H] :
  calc x = y := hy' x hx _ = 1 := (hy' 1 H.one_mem).symm - lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) [Fintype H] [Fintype K] + lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) (h : (Nat.card H).Coprime (Nat.card K)) : H ⊓ K = ⊥ := by have D₁ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card H := card_dvd_of_le inf_le_left have D₂ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card K := card_dvd_of_le inf_le_right
- @@ -127,7 +127,7 @@ open MonoidHom
  #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right - lemma aux_card_eq [Fintype G] (h' : Nat.card G = Nat.card H * Nat.card K) : + lemma aux_card_eq [Finite G] (h' : Nat.card G = Nat.card H * Nat.card K) : Nat.card (G ⧸ H) = Nat.card K := by have := calc Nat.card (G ⧸ H) * Nat.card H = Nat.card G := by rw [← H.index_eq_card, H.index_mul_card]
MIL/C08_Groups_and_Rings/solutions/Solutions_S02_Rings.lean
- @@ -1,7 +1,7 @@
  import Mathlib.RingTheory.Ideal.QuotientOperations import Mathlib.RingTheory.Localization.Basic import Mathlib.RingTheory.DedekindDomain.Ideal - import Mathlib.Analysis.Complex.Polynomial + import Mathlib.Analysis.Complex.Polynomial.Basic import Mathlib.Data.ZMod.Quotient import MIL.Common
MIL/C09_Topology/S02_Metric_Spaces.lean
- @@ -1,6 +1,6 @@
  import MIL.Common import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus + import Mathlib.Analysis.Normed.Operator.BanachSteinhaus open Set Filter open Topology Filter
MIL/C09_Topology/S03_Topological_Spaces.lean
- @@ -1,6 +1,6 @@
  import MIL.Common import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus + import Mathlib.Analysis.Normed.Operator.BanachSteinhaus open Set Filter Topology
MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean
- @@ -1,6 +1,6 @@
  import MIL.Common import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus + import Mathlib.Analysis.Normed.Operator.BanachSteinhaus open Set Filter open Topology Filter
MIL/C09_Topology/solutions/Solutions_S03_Topological_Spaces.lean
- @@ -1,6 +1,6 @@
  import MIL.Common import Mathlib.Topology.Instances.Real - import Mathlib.Analysis.NormedSpace.BanachSteinhaus + import Mathlib.Analysis.Normed.Operator.BanachSteinhaus open Set Filter Topology
MIL/C10_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean
- @@ -1,6 +1,6 @@
  import MIL.Common - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Normed.Operator.BanachSteinhaus + import Mathlib.Analysis.Normed.Module.FiniteDimension import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv import Mathlib.Analysis.Calculus.ContDiff.RCLike import Mathlib.Analysis.Calculus.FDeriv.Prod
MIL/C10_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean
- @@ -1,6 +1,6 @@
  import MIL.Common - import Mathlib.Analysis.NormedSpace.BanachSteinhaus - import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Normed.Operator.BanachSteinhaus + import Mathlib.Analysis.Normed.Module.FiniteDimension import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv import Mathlib.Analysis.Calculus.ContDiff.RCLike import Mathlib.Analysis.Calculus.FDeriv.Prod
MIL/C11_Integration_and_Measure_Theory/S02_Measure_Theory.lean
- @@ -1,5 +1,5 @@
  import MIL.Common - import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Normed.Module.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian import Mathlib.MeasureTheory.Integral.Bochner
MIL/C11_Integration_and_Measure_Theory/S03_Integration.lean
- @@ -1,5 +1,5 @@
  import MIL.Common - import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Normed.Module.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian import Mathlib.MeasureTheory.Integral.Bochner
MIL/C11_Integration_and_Measure_Theory/solutions/Solutions_S02_Measure_Theory.lean
- @@ -1,5 +1,5 @@
  import MIL.Common - import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Normed.Module.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian import Mathlib.MeasureTheory.Integral.Bochner
MIL/C11_Integration_and_Measure_Theory/solutions/Solutions_S03_Integration.lean
- @@ -1,5 +1,5 @@
  import MIL.Common - import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Normed.Module.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian import Mathlib.MeasureTheory.Integral.Bochner
html/C01_Introduction.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>1. Introduction — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" />
- @@ -90,9 +92,9 @@
 <div itemprop="articleBody"> <section id="introduction"> - <h1>1. Introduction<a class="headerlink" href="#introduction" title="Permalink to this heading"></a></h1> + <h1>1. Introduction<a class="headerlink" href="#introduction" title="Link to this heading"></a></h1> <section id="getting-started"> - <h2>1.1. Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this heading"></a></h2> + <h2>1.1. Getting Started<a class="headerlink" href="#getting-started" title="Link to this heading"></a></h2> The goal of this book is to teach you to formalize mathematics using the Lean 4 interactive proof assistant. It assumes that you know some mathematics, but it does not require much.
- @@ -140,10 +142,10 @@ your own Lean files as well.</li>
 At that point, you can open the textbook in a side panel in VS Code as follows: <ol class="arabic simple"> <li>Type <code class="docutils literal notranslate">ctrl-shift-P</code> (<code class="docutils literal notranslate">command-shift-P</code> in macOS).</li> - <li>Type <code class="docutils literal notranslate">Lean 4: Open Documentation View</code> in the bar that appears, and then + <li>Type <code class="docutils literal notranslate">Lean 4: Docs: Show Documentation Resources</code> in the bar that appears, and then press return. (You can press return to select it as soon as it is highlighted in the menu.)</li> - <li>In the window that opens, click on <code class="docutils literal notranslate">Open documentation of current project</code>.</li> + <li>In the window that opens, click on <code class="docutils literal notranslate">Mathematics in Lean</code>.</li> </ol> Alternatively, you can run Lean and VS Code in the cloud, using <a class="reference external" href="https://gitpod.io/">Gitpod</a>.
- @@ -179,7 +181,7 @@ You can always compare your solutions to the ones in the <code class="docutils l
 folder associated with each section. </section> <section id="overview"> - <h2>1.2. Overview<a class="headerlink" href="#overview" title="Permalink to this heading"></a></h2> + <h2>1.2. Overview<a class="headerlink" href="#overview" title="Link to this heading"></a></h2> Put simply, Lean is a tool for building complex expressions in a formal language known as dependent type theory. Every expression has a type, and you can use the <cite>#check</cite> command to
- @@ -342,15 +344,16 @@ Your life may never be the same.
 infrastructure for running this tutorial in VS Code, and to Scott Morrison and Mario Carneiro for help porting it from Lean 4. We are also grateful for help and corrections from - Takeshi Abe, Julian Berman, Alex Best, - Bulwi Cha, Bryan Gin-ge Chen, Steven Clontz, Mauricio Collaris, Johan Commelin, Mark Czubin, - Alexandru Duca, Denis Gorbachev, Winston de Greef, - Mathieu Guay-Paquet, Julian Külshammer, + Takeshi Abe, Julian Berman, Alex Best, Thomas Browning, + Bulwi Cha, Hanson Char, Bryan Gin-ge Chen, Steven Clontz, Mauricio Collaris, Johan Commelin, Mark Czubin, + Alexandru Duca, Pierpaolo Frasa, Denis Gorbachev, Winston de Greef, Marc Huisinga, + Mathieu Guay-Paquet, Julian Külshammer, Victor Liu, Jimmy Lu, Martin C. Martin, - Giovanni Mascellani, Isaiah Mindich, Hunter Monroe, Pietro Monticone, Oliver Nash, - Bartosz Piotrowski, Nicolas Rolland, Keith Rush, Guilherme Silva, - Pedro Sánchez Terraf, - Floris van Doorn, and Eric Wieser. + Giovanni Mascellani, John McDowell, Isaiah Mindich, Hunter Monroe, Pietro Monticone, + Oliver Nash, Emanuelle Natale, Pim Otte, + Bartosz Piotrowski, Nicolas Rolland, Keith Rush, Yannick Seurin, Guilherme Silva, + Pedro Sánchez Terraf, Matthew Toohey, + Floris van Doorn, Eric Wieser, and others. Our work has been partially supported by the Hoskinson Center for Formal Mathematics. </section>
html/C02_Basics.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>2. Basics — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -94,13 +96,13 @@
 <div itemprop="articleBody"> <section id="basics"> - <h1>2. Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h1> + <h1>2. Basics<a class="headerlink" href="#basics" title="Link to this heading"></a></h1> This chapter is designed to introduce you to the nuts and bolts of mathematical reasoning in Lean: calculating, applying lemmas and theorems, and reasoning about generic structures. <section id="calculating"> - <h2>2.1. Calculating<a class="headerlink" href="#calculating" title="Permalink to this heading"></a></h2> + <h2>2.1. Calculating<a class="headerlink" href="#calculating" title="Link to this heading"></a></h2> We generally learn to carry out mathematical calculations without thinking of them as proofs. But when we justify each step in a calculation,
- @@ -391,7 +393,7 @@ occurrence of <code class="docutils literal notranslate">a</sp
 </div> </section> <section id="proving-identities-in-algebraic-structures"> - <h2>2.2. Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this heading"></a></h2> + <h2>2.2. Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Link to this heading"></a></h2> Mathematically, a ring consists of a collection of objects, $R$, operations $+$ $\times$, and constants $0$ and $1$, and an operation $x \mapsto -x$ such that:
- @@ -711,7 +713,7 @@ but also for the convenience of using a shorter name for the
 tactic that deals with commutative rings, since it is used more often. </section> <section id="using-theorems-and-lemmas"> - <h2>2.3. Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this heading"></a></h2> + <h2>2.3. Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Link to this heading"></a></h2> Rewriting is great for proving equations, but what about other sorts of theorems? For example, how can we prove an inequality,
- @@ -978,7 +980,7 @@ to two goals; see <a class="reference internal" href="C03_Logic.html#conjunction
 You are well on your way to becoming a master formalizer. </section> <section id="more-examples-using-apply-and-rw"> - <h2>2.4. More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Permalink to this heading"></a></h2> + <h2>2.4. More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Link to this heading"></a></h2> The <code class="docutils literal notranslate">min</code> function on the real numbers is uniquely characterized by the following three facts: <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (min_le_left a b : min a b ≤ a)
- @@ -1179,7 +1181,7 @@ You can use <code class="docutils literal notranslate">_root_.
 either one will work. </section> <section id="proving-facts-about-algebraic-structures"> - <h2>2.5. Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this heading"></a></h2> + <h2>2.5. Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Link to this heading"></a></h2> In <a class="reference internal" href="#proving-identities-in-algebraic-structures">Section 2.2</a>, we saw that many common identities governing the real numbers hold in more general classes of algebraic structures,
- @@ -1285,7 +1287,7 @@ together with <code class="docutils literal notranslate">le_re
 Using <code class="docutils literal notranslate">apply le_trans</code> when seeing a goal <code class="docutils literal notranslate">x ≤ z</code> is not a great idea. Indeed Lean has no way to guess which intermediate element <code class="docutils literal notranslate">y</code> we want to use. - So <code class="docutils literal notranslate">apply le_trans</code> produces three goals that look like``x ≤ ?a``, <code class="docutils literal notranslate">?a ≤ z</code> + So <code class="docutils literal notranslate">apply le_trans</code> produces three goals that look like <code class="docutils literal notranslate">x ≤ ?a</code>, <code class="docutils literal notranslate">?a ≤ z</code> and <code class="docutils literal notranslate">α</code> where <code class="docutils literal notranslate">?a</code> (probably with a more complicated auto-generated name) stands for the mysterious <code class="docutils literal notranslate">y</code>. The last goal, with type <code class="docutils literal notranslate">α</code>, is to provide the value of <code class="docutils literal notranslate">y</code>.
html/C03_Logic.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>3. Logic — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -95,7 +97,7 @@
 <div itemprop="articleBody"> <section id="logic"> - <h1>3. Logic<a class="headerlink" href="#logic" title="Permalink to this heading"></a></h1> + <h1>3. Logic<a class="headerlink" href="#logic" title="Link to this heading"></a></h1> In the last chapter, we dealt with equations, inequalities, and basic mathematical statements like “$x$ divides $y$.”
- @@ -106,7 +108,7 @@ using logical terms like “and,” “or,” “not,”
 In this chapter, we show you how to work with statements that are built up in this way. <section id="implication-and-the-universal-quantifier"> - <h2>3.1. Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this heading"></a></h2> + <h2>3.1. Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Link to this heading"></a></h2> Consider the statement after the <code class="docutils literal notranslate">#check</code>: <div class="highlight-lean notranslate"><div class="highlight"><pre>#check ∀ x : ℝ, 0 ≤ x → |x| = x </pre></div>
- @@ -496,7 +498,7 @@ a lemma name.
 </div> </section> <section id="the-existential-quantifier"> - <h2>3.2. The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this heading"></a></h2> + <h2>3.2. The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Link to this heading"></a></h2> The existential quantifier, which can be entered as <code class="docutils literal notranslate">\ex</code> in VS Code, is used to represent the phrase “there exists.” The formal expression <code class="docutils literal notranslate">∃ x : ℝ, 2 < x ∧ x < 3</code> in Lean says
- @@ -817,7 +819,7 @@ the composition of surjective functions is surjective.
 </div> </section> <section id="negation"> - <h2>3.3. Negation<a class="headerlink" href="#negation" title="Permalink to this heading"></a></h2> + <h2>3.3. Negation<a class="headerlink" href="#negation" title="Link to this heading"></a></h2> The symbol <code class="docutils literal notranslate">¬</code> is meant to express negation, so <code class="docutils literal notranslate">¬ x < y</code> says that <code class="docutils literal notranslate">x</code> is not less than <code class="docutils literal notranslate">y</code>, <code class="docutils literal notranslate">¬ x = y</code> (or, equivalently, <code class="docutils literal notranslate">x ≠ y</code>) says that
- @@ -1082,7 +1084,7 @@ such as a pair of the form <code class="docutils literal notranslate">linarith</code> also works. </section> <section id="conjunction-and-iff"> - <h2>3.4. Conjunction and Iff<a class="headerlink" href="#conjunction-and-iff" title="Permalink to this heading"></a></h2> + <h2>3.4. Conjunction and Iff<a class="headerlink" href="#conjunction-and-iff" title="Link to this heading"></a></h2> You have already seen that the conjunction symbol, <code class="docutils literal notranslate">∧</code>, is used to express “and.” The <code class="docutils literal notranslate">constructor</code> tactic allows you to prove a statement of
- @@ -1341,7 +1343,7 @@ to be instantiated to different values.
 </div> </section> <section id="disjunction"> - <h2>3.5. Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this heading"></a></h2> + <h2>3.5. Disjunction<a class="headerlink" href="#disjunction" title="Link to this heading"></a></h2> The canonical way to prove a disjunction <code class="docutils literal notranslate">A ∨ B</code> is to prove <code class="docutils literal notranslate">A</code> or to prove <code class="docutils literal notranslate">B</code>. The <code class="docutils literal notranslate">left</code> tactic chooses <code class="docutils literal notranslate">A</code>,
- @@ -1583,7 +1585,7 @@ using <code class="docutils literal notranslate">by_cases</spa
 </div> </section> <section id="sequences-and-convergence"> - <h2>3.6. Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this heading"></a></h2> + <h2>3.6. Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Link to this heading"></a></h2> We now have enough skills at our disposal to do some real mathematics. In Lean, we can represent a sequence $s_0, s_1, s_2, \ldots$ of real numbers as a function <code class="docutils literal notranslate">s : ℕ → ℝ</code>.
html/C04_Sets_and_Functions.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>4. Sets and Functions — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -92,7 +94,7 @@
 <div itemprop="articleBody"> <section id="sets-and-functions"> - <h1>4. Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this heading"></a></h1> + <h1>4. Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Link to this heading"></a></h1> The vocabulary of sets, relations, and functions provides a uniform language for carrying out constructions in all the branches of mathematics.
- @@ -125,7 +127,7 @@ from real numbers to real numbers.
 The distinction between types and sets takes some getting used to, but this chapter will take you through the essentials. <section id="sets"> - <h2>4.1. Sets<a class="headerlink" href="#sets" title="Permalink to this heading"></a></h2> + <h2>4.1. Sets<a class="headerlink" href="#sets" title="Link to this heading"></a></h2> If <code class="docutils literal notranslate">α</code> is any type, the type <code class="docutils literal notranslate">Set α</code> consists of sets of elements of <code class="docutils literal notranslate">α</code>. This type supports the usual set-theoretic operations and relations.
- @@ -554,7 +556,7 @@ and intersection.
 <code class="docutils literal notranslate">sUnion_eq_biUnion</code> and <code class="docutils literal notranslate">sInter_eq_biInter</code>. </section> <section id="functions"> - <h2>4.2. Functions<a class="headerlink" href="#functions" title="Permalink to this heading"></a></h2> + <h2>4.2. Functions<a class="headerlink" href="#functions" title="Link to this heading"></a></h2> If <code class="docutils literal notranslate">f : α → β</code> is a function and <code class="docutils literal notranslate">p</code> is a set of elements of type <code class="docutils literal notranslate">β</code>, the library defines <code class="docutils literal notranslate">preimage f p</code>, written <code class="docutils literal notranslate">f ⁻¹' p</code>,
- @@ -862,7 +864,7 @@ and then fill in the two lines that are missing.
 </div> </section> <section id="the-schroder-bernstein-theorem"> - <h2>4.3. The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this heading"></a></h2> + <h2>4.3. The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Link to this heading"></a></h2> We close this chapter with an elementary but nontrivial theorem of set theory. Let $\alpha$ and $\beta$ be sets. (In our formalization, they will actually be types.)
html/C05_Elementary_Number_Theory.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>5. Elementary Number Theory — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -92,14 +94,14 @@
 <div itemprop="articleBody"> <section id="elementary-number-theory"> - <h1>5. Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Permalink to this heading"></a></h1> + <h1>5. Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Link to this heading"></a></h1> In this chapter, we show you how to formalize some elementary results in number theory. As we deal with more substantive mathematical content, the proofs will get longer and more involved, building on the skills you have already mastered. <section id="irrational-roots"> - <h2>5.1. Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this heading"></a></h2> + <h2>5.1. Irrational Roots<a class="headerlink" href="#irrational-roots" title="Link to this heading"></a></h2> Let’s start with a fact known to the ancient Greeks, namely, that the square root of 2 is irrational. If we suppose otherwise,
- @@ -264,17 +266,17 @@ The formalization below confirms that this assumption is sufficient.
 The unique factorization theorem says that any natural number other than zero can be written as the product of primes in a unique way. Mathlib contains a formal version of this, expressed in terms of a function - <code class="docutils literal notranslate">Nat.factors</code>, which returns the list of + <code class="docutils literal notranslate">Nat.primeFactorsList</code>, which returns the list of prime factors of a number in nondecreasing order. - The library proves that all the elements of <code class="docutils literal notranslate">Nat.factors n</code> + The library proves that all the elements of <code class="docutils literal notranslate">Nat.primeFactorsList n</code> are prime, that any <code class="docutils literal notranslate">n</code> greater than zero is equal to the product of its factors, and that if <code class="docutils literal notranslate">n</code> is equal to the product of another list of prime numbers, - then that list is a permutation of <code class="docutils literal notranslate">Nat.factors n</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Nat.factors - #check Nat.prime_of_mem_factors - #check Nat.prod_factors - #check Nat.factors_unique + then that list is a permutation of <code class="docutils literal notranslate">Nat.primeFactorsList n</code>. + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check Nat.primeFactorsList + #check Nat.prime_of_mem_primeFactorsList + #check Nat.prod_primeFactorsList + #check Nat.primeFactorsList_unique </pre></div> </div> You can browse these theorems and others nearby, even though we have not
- @@ -402,7 +404,7 @@ In the next chapter, we will begin to develop the means to
 appreciate the way that Lean supports this sort of generality. </section> <section id="induction-and-recursion"> - <h2>5.2. Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this heading"></a></h2> + <h2>5.2. Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Link to this heading"></a></h2> The set of natural numbers $\mathbb{N} = \{ 0, 1, 2, \ldots \}$ is not only fundamentally important in its own right, but also a plays a central role in the construction of new mathematical objects.
- @@ -410,7 +412,7 @@ Lean’s foundation allows us to declare inductive types,
 which are types generated inductively by a given list of constructors. In Lean, the natural numbers are declared as follows. - <div class="highlight-lean notranslate"><div class="highlight"><pre>inductive Nat + <div class="highlight-lean notranslate"><div class="highlight"><pre>inductive Nat where | zero : Nat | succ (n : Nat) : Nat </pre></div>
- @@ -527,7 +529,7 @@ or <code class="docutils literal notranslate">pow_succ'
 Induction is often used to prove identities involving finite sums and products. Mathlib defines the expressions <code class="docutils literal notranslate">Finset.sum s f</code> where - <code class="docutils literal notranslate">s : Finset α</code> if a finite set of elements of the type <code class="docutils literal notranslate">α</code> and + <code class="docutils literal notranslate">s : Finset α</code> is a finite set of elements of the type <code class="docutils literal notranslate">α</code> and <code class="docutils literal notranslate">f</code> is a function defined on <code class="docutils literal notranslate">α</code>. The codomain of <code class="docutils literal notranslate">f</code> can be any type that supports a commutative, associative addition operation with a zero element.
- @@ -562,7 +564,7 @@ less than <code class="docutils literal notranslate">n<
 </pre></div> </div> The facts <code class="docutils literal notranslate">Finset.sum_range_zero</code> and <code class="docutils literal notranslate">Finset.sum_range_succ</code> - provide a recursive description summation up to $n$, + provide a recursive description of summation up to $n$, and similarly for products. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℕ → ℕ) : ∑ x in range 0, f x = 0 := Finset.sum_range_zero f
- @@ -582,8 +584,8 @@ you can replace the proofs by <code class="docutils literal notranslate">The following expresses the factorial function that we defined as a product. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := by induction' n with n ih - · rw [fac, prod_range_zero] - rw [fac, ih, prod_range_succ, mul_comm] + · simp [fac, prod_range_zero] + simp [fac, ih, prod_range_succ, mul_comm] </pre></div> </div> The fact that we include <code class="docutils literal notranslate">mul_comm</code> as a simplification rule deserves
- @@ -662,7 +664,7 @@ Remember that truncated subtraction cuts off at zero.
 To define that, it is useful to define a predecessor function, <code class="docutils literal notranslate">pred</code>, that subtracts one from any nonzero number and fixes zero. The function <code class="docutils literal notranslate">pred</code> can be defined by a simple instance of recursion. - <div class="highlight-lean notranslate"><div class="highlight"><pre>inductive MyNat + <div class="highlight-lean notranslate"><div class="highlight"><pre>inductive MyNat where | zero : MyNat | succ : MyNat → MyNat
- @@ -708,7 +710,7 @@ The function <code class="docutils literal notranslate">pred</
 </div> </section> <section id="infinitely-many-primes"> - <h2>5.3. Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this heading"></a></h2> + <h2>5.3. Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Link to this heading"></a></h2> Let us continue our exploration of induction and recursion with another mathematical standard: a proof that there are infinitely many primes. One way to formulate this is as the statement that
- @@ -732,7 +734,7 @@ successor function respects the ordering on the natural numbers.
 cases m; contradiction case succ m => cases m; contradiction - repeat' apply Nat.succ_le_succ + repeat apply Nat.succ_le_succ apply zero_le </pre></div> </div>
- @@ -749,7 +751,7 @@ Remember that you can hover over it to see its documentation.
 </div> Recall that the semicolon after <code class="docutils literal notranslate">interval_cases m</code> means that the next tactic is applied to each of the cases that it generates. - Yet another option is to use the tactic, <code class="docutils literal notranslate">decide</code>, which tries + Yet another option is to use the tactic <code class="docutils literal notranslate">decide</code>, which tries to find a decision procedure to solve the problem. Lean knows that you can decide the truth value of a statement that begins with a bounded quantifier <code class="docutils literal notranslate">∀ x, x < n → ...</code> or <code class="docutils literal notranslate">∃ x, x < n ∧ ...</code>
- @@ -783,10 +785,7 @@ if $n$ is prime, we’re
 then by one of the characterizations of what it means to be a prime number, it has a nontrivial factor, $m$, and we can apply the inductive hypothesis to that. - Step through the next proof to see how that plays out. - The line <code class="docutils literal notranslate">dsimp at ih</code> simplifies the expression of the - inductive hypothesis to make it more readable. - The proof still works if you delete that line. + Step through the next proof to see how that plays out. <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by by_cases np : n.Prime · use n, np
- @@ -851,7 +850,7 @@ most equivalences involving finsets do not hold definitionally,
 so they need to be expanded manually using equivalences like <code class="docutils literal notranslate">Finset.subset_iff</code>, <code class="docutils literal notranslate">Finset.mem_union</code>, <code class="docutils literal notranslate">Finset.mem_inter</code>, and <code class="docutils literal notranslate">Finset.mem_sdiff</code>. The <code class="docutils literal notranslate">ext</code> tactic can still be used - to reduce show that two finite sets are equal by showing + to show that two finite sets are equal by showing that every element of one is an element of the other. <div class="highlight-lean notranslate"><div class="highlight"><pre>open Finset
- @@ -910,7 +909,7 @@ be able to prove using the theorem <code class="docutils literal notranslate"><s
 </pre></div> </div> We can use this lemma to show that if a prime <code class="docutils literal notranslate">p</code> divides a product of a finite - set of primes, then it divides one of them. + set of primes, then it is equal to one of them. Mathlib provides a useful principle of induction on finite sets: to show that a property holds of an arbitrary finite set <code class="docutils literal notranslate">s</code>, show that it holds of the empty set, and show that it is preserved
- @@ -960,7 +959,7 @@ You can use <code class="docutils literal notranslate">Finset.
 <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by intro s by_contra h - push_neg at h + push_neg at h set s' := s.filter Nat.Prime with s'_def have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by intro n
- @@ -1013,7 +1012,7 @@ where <code class="docutils literal notranslate">id</co
 shows that there are infinitely many primes congruent to 3 modulo 4. The argument goes as follows. First, notice that if the product of two numbers $m$ and $n$ - is equal to 3 modulo 4, then one of the two numbers is congruent to three modulo 4. + is equal to 3 modulo 4, then one of the two numbers is congruent to 3 modulo 4. After all, both have to be odd, and if they are both congruent to 1 modulo 4, so is their product. We can use this observation to show that if some number
- @@ -1082,7 +1081,7 @@ same property.
 · use n induction' n using Nat.strong_induction_on with n ih rw [Nat.prime_def_lt] at np - push_neg at np + push_neg at np rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ have mge2 : 2 ≤ m := by apply two_le _ mne1
- @@ -1116,7 +1115,7 @@ Our solution uses <code class="docutils literal notranslate">N
 along the way. <div class="highlight-lean notranslate"><div class="highlight"><pre>theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by by_contra h - push_neg at h + push_neg at h rcases h with ⟨n, hn⟩ have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by apply ex_finset_of_bounded
html/C06_Structures.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>6. Structures — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -92,7 +94,7 @@
 <div itemprop="articleBody"> <section id="structures"> - <h1>6. Structures<a class="headerlink" href="#structures" title="Permalink to this heading"></a></h1> + <h1>6. Structures<a class="headerlink" href="#structures" title="Link to this heading"></a></h1> Modern mathematics makes essential use of algebraic structures, which encapsulate patterns that can be instantiated in
- @@ -112,14 +114,14 @@ algebraic structures on your own.
 For more technical detail, you can consult <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, and a paper by Anne Baanen, <a class="reference external" href="https://arxiv.org/abs/2202.01629">Use and abuse of instance parameters in the Lean mathematical library</a>. <section id="defining-structures"> - <h2>6.1. Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this heading"></a></h2> + <h2>6.1. Defining structures<a class="headerlink" href="#defining-structures" title="Link to this heading"></a></h2> In the broadest sense of the term, a structure is a specification of a collection of data, possibly with constraints that the data is required to satisfy. An instance of the structure is a particular bundle of data satisfying the constraints. For example, we can specify that a point is a tuple of three real numbers: - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] structure Point where x : ℝ y : ℝ
- @@ -473,7 +475,7 @@ weaving structures together into a rich, interconnected hierarchy,
 and for managing the interactions between them. </section> <section id="algebraic-structures"> - <h2>6.2. Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this heading"></a></h2> + <h2>6.2. Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Link to this heading"></a></h2> To clarify what we mean by the phrase algebraic structure, it will help to consider some examples. <ol class="arabic simple">
- @@ -739,7 +741,7 @@ and define the <code class="docutils literal notranslate">AddG
 <div class="highlight-lean notranslate"><div class="highlight"><pre>structure AddGroup₁ (α : Type*) where (add : α → α → α) -- fill in the rest - @[ext] + @[ext] structure Point where x : ℝ y : ℝ
- @@ -1022,7 +1024,7 @@ When used wisely, however, class inference is a powerful tool.
 It is what makes algebraic reasoning possible in Lean. </section> <section id="building-the-gaussian-integers"> - <h2>6.3. Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this heading"></a></h2> + <h2>6.3. Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Link to this heading"></a></h2> We will now illustrate the use of the algebraic hierarchy in Lean by building an important mathematical object, the Gaussian integers, and showing that it is a Euclidean domain. In other words, according to
- @@ -1034,7 +1036,7 @@ But rather than define them as a subset of the complex numbers, our goal
 here is to define them as a data type in their own right. We do this by representing a Gaussian integer as a pair of integers, which we think of as the real and imaginary parts. - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] structure GaussInt where re : ℤ im : ℤ
- @@ -1093,43 +1095,43 @@ to use with <code class="docutils literal notranslate">simp</s
 </div> It is also useful to name the rules that compute the real and imaginary parts, and to declare them to the simplifier. - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[simp] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[simp] theorem zero_re : (0 : GaussInt).re = 0 := rfl - @[simp] + @[simp] theorem zero_im : (0 : GaussInt).im = 0 := rfl - @[simp] + @[simp] theorem one_re : (1 : GaussInt).re = 1 := rfl - @[simp] + @[simp] theorem one_im : (1 : GaussInt).im = 0 := rfl - @[simp] + @[simp] theorem add_re (x y : GaussInt) : (x + y).re = x.re + y.re := rfl - @[simp] + @[simp] theorem add_im (x y : GaussInt) : (x + y).im = x.im + y.im := rfl - @[simp] + @[simp] theorem neg_re (x : GaussInt) : (-x).re = -x.re := rfl - @[simp] + @[simp] theorem neg_im (x : GaussInt) : (-x).im = -x.im := rfl - @[simp] + @[simp] theorem mul_re (x y : GaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl - @[simp] + @[simp] theorem mul_im (x y : GaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl </pre></div>
- @@ -1366,7 +1368,7 @@ The proofs are all short.
 <div class="highlight-lean notranslate"><div class="highlight"><pre>def norm (x : GaussInt) := x.re ^ 2 + x.im ^ 2 - @[simp] + @[simp] theorem norm_nonneg (x : GaussInt) : 0 ≤ norm x := by sorry theorem norm_eq_zero (x : GaussInt) : norm x = 0 ↔ x = 0 := by
- @@ -1381,11 +1383,11 @@ The proofs are all short.
 <div class="highlight-lean notranslate"><div class="highlight"><pre>def conj (x : GaussInt) : GaussInt := ⟨x.re, -x.im⟩ - @[simp] + @[simp] theorem conj_re (x : GaussInt) : (conj x).re = x.re := rfl - @[simp] + @[simp] theorem conj_im (x : GaussInt) : (conj x).im = -x.im := rfl
html/C07_Hierarchies.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>7. Hierarchies — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" />
- @@ -91,7 +93,7 @@
 <div itemprop="articleBody"> <section id="hierarchies"> - <h1>7. Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this heading"></a></h1> + <h1>7. Hierarchies<a class="headerlink" href="#hierarchies" title="Link to this heading"></a></h1> We have seen in <a class="reference internal" href="C06_Structures.html#structures">Chapter 6</a> how to define the class of groups and build instances of this class, and then how to build an instance of the commutative ring class. But of course there is a hierarchy here: a
- @@ -108,7 +110,7 @@ so we will used indices to distinguish our version. For instance we will have <c
 as our version of <code class="docutils literal notranslate">Ring</code>. Since we will gradually explain more powerful ways of formalizing structures, those indices will sometimes grow beyond one. <section id="basics"> - <h2>7.1. Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h2> + <h2>7.1. Basics<a class="headerlink" href="#basics" title="Link to this heading"></a></h2> At the very bottom of all hierarchies in Lean, we find data-carrying classes. The following class records that the given type <code class="docutils literal notranslate">α</code> is endowed with a distinguished element called <code class="docutils literal notranslate">one</code>. At this stage, it has no property at all.
- @@ -128,7 +130,7 @@ attribute, ie writing <code class="docutils literal notranslate">One₁ α</code> appears as an instance-implicit argument in its own fields. Compare: <div class="highlight-lean notranslate"><div class="highlight"><pre>#check One₁.one -- One₁.one {α : Type} [self : One₁ α] : α - @[class] structure One₂ (α : Type) where + @[class] structure One₂ (α : Type) where /-- The element one -/ one : α
- @@ -162,7 +164,7 @@ or real numbers.
 with the builtin notation for <code class="docutils literal notranslate">1</code>, we will use <code class="docutils literal notranslate">𝟙</code>. This is achieved by the following command where the first line tells Lean to use the documentation of <code class="docutils literal notranslate">One₁.one</code> as documentation for the symbol <code class="docutils literal notranslate">𝟙</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[inherit_doc] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[inherit_doc] notation "𝟙" => One₁.one example {α : Type} [One₁ α] : α := 𝟙
- @@ -280,7 +282,7 @@ we define a new data-carrying class, and then give it some notation.
 /-- The inversion function -/ inv : α → α - @[inherit_doc] + @[inherit_doc] postfix:max "⁻¹" => Inv₁.inv class Group₁ (G : Type) extends Monoid₁ G, Inv₁ G where
- @@ -333,14 +335,14 @@ used the <code class="docutils literal notranslate">whatsnew</
 /-- Addition is associative -/ add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) - @[to_additive AddSemigroup₃] + @[to_additive AddSemigroup₃] class Semigroup₃ (α : Type) extends Mul α where /-- Multiplication is associative -/ mul_assoc₃ : ∀ a b c : α, a * b * c = a * (b * c) class AddMonoid₃ (α : Type) extends AddSemigroup₃ α, AddZeroClass α - @[to_additive AddMonoid₃] + @[to_additive AddMonoid₃] class Monoid₃ (α : Type) extends Semigroup₃ α, MulOneClass α attribute [to_additive existing] Monoid₃.toMulOneClass
- @@ -349,7 +351,7 @@ used the <code class="docutils literal notranslate">whatsnew</
 export AddSemigroup₃ (add_assoc₃) whatsnew in - @[to_additive] + @[to_additive] lemma left_inv_eq_right_inv' {M : Type} [Monoid₃ M] {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c := by rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b]
- @@ -361,19 +363,19 @@ groups, and then define rings.
 <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddCommSemigroup₃ (α : Type) extends AddSemigroup₃ α where add_comm : ∀ a b : α, a + b = b + a - @[to_additive AddCommSemigroup₃] + @[to_additive AddCommSemigroup₃] class CommSemigroup₃ (α : Type) extends Semigroup₃ α where mul_comm : ∀ a b : α, a * b = b * a class AddCommMonoid₃ (α : Type) extends AddMonoid₃ α, AddCommSemigroup₃ α - @[to_additive AddCommMonoid₃] + @[to_additive AddCommMonoid₃] class CommMonoid₃ (α : Type) extends Monoid₃ α, CommSemigroup₃ α class AddGroup₃ (G : Type) extends AddMonoid₃ G, Neg G where neg_add : ∀ a : G, -a + a = 0 - @[to_additive AddGroup₃] + @[to_additive AddGroup₃] class Group₃ (G : Type) extends Monoid₃ G, Inv G where inv_mul : ∀ a : G, a⁻¹ * a = 1 </pre></div>
- @@ -384,28 +386,28 @@ groups, and then define rings.
 </div> Then we need to repeat ourselves a bit since we switch to standard notations, but at least <code class="docutils literal notranslate">to_additive</code> does the work of translating from the multiplicative notation to the additive one. - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive] lemma inv_eq_of_mul [Group₃ G] {a b : G} (h : a * b = 1) : a⁻¹ = b := sorry </pre></div> </div> Note that <code class="docutils literal notranslate">to_additive</code> can be asked to tag a lemma with <code class="docutils literal notranslate">simp</code> and propagate that attribute to the additive version as follows. - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive (attr := simp)] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[to_additive (attr := simp)] lemma Group₃.mul_inv {G : Type} [Group₃ G] {a : G} : a * a⁻¹ = 1 := by sorry - @[to_additive] + @[to_additive] lemma mul_left_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : a * b = a * c) : b = c := by sorry - @[to_additive] + @[to_additive] lemma mul_right_cancel₃ {G : Type} [Group₃ G] {a b c : G} (h : b*a = c*a) : b = c := by sorry class AddCommGroup₃ (G : Type) extends AddGroup₃ G, AddCommMonoid₃ G - @[to_additive AddCommGroup₃] + @[to_additive AddCommGroup₃] class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G </pre></div> </div>
- @@ -457,7 +459,7 @@ such that <code class="docutils literal notranslate">∀</
 /-- The Less-or-Equal relation. -/ le : α → α → Prop - @[inherit_doc] infix:50 " ≤₁ " => LE₁.le + @[inherit_doc] infix:50 " ≤₁ " => LE₁.le class Preorder₁ (α : Type)
- @@ -635,7 +637,7 @@ that every preorder comes with a <code class="docutils literal notranslate"><spa
 -/ </section> <section id="morphisms"> - <h2>7.2. Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this heading"></a></h2> + <h2>7.2. Morphisms<a class="headerlink" href="#morphisms" title="Link to this heading"></a></h2> So far in this chapter, we discussed how to create a hierarchy of mathematical structures. But defining structures is not really completed until we have morphisms. There are two main approaches here. The most obvious one is to define a predicate on functions.
- @@ -672,7 +674,7 @@ is really a function that happens to be continuous. This is one reason why Mathl
 We still have bundles continuous functions, which are convenient for instance to put a topology on a space of continuous functions, but they are not the primary tool to work with continuity. By contrast, morphisms between monoids (or other algebraic structures) are bundled as in: - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where toFun : G → H map_one : toFun 1 = 1
- @@ -695,7 +697,7 @@ make sure it is displayed almost invisibly in the tactic state, simply by a <cod
 </pre></div> </div> We can do the same with other kind of morphisms until we reach ring morphisms. - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] structure AddMonoidHom₁ (G H : Type) [AddMonoid G] [AddMonoid H] where toFun : G → H map_zero : toFun 0 = 0
- @@ -706,7 +708,7 @@ make sure it is displayed almost invisibly in the tactic state, simply by a <cod
 attribute [coe] AddMonoidHom₁.toFun - @[ext] + @[ext] structure RingHom₁ (R S : Type) [Ring R] [Ring S] extends MonoidHom₁ R S, AddMonoidHom₁ R S </pre></div> </div>
- @@ -817,12 +819,12 @@ ordered types, and then order preserving monoid morphisms. This is for training
 Like continuous functions, order preserving functions are primarily unbundled in Mathlib where they are defined by the <code class="docutils literal notranslate">Monotone</code> predicate. Of course you need to complete the class definitions below. - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] structure OrderPresHom (α β : Type) [LE α] [LE β] where toFun : α → β le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a' - @[ext] + @[ext] structure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends MonoidHom₁ M N, OrderPresHom M N
- @@ -840,7 +842,7 @@ definitions below.
 </div> </section> <section id="sub-objects"> - <h2>7.3. Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this heading"></a></h2> + <h2>7.3. Sub-objects<a class="headerlink" href="#sub-objects" title="Link to this heading"></a></h2> After defining some algebraic structure and its morphisms, the next step is to consider sets that inherit this algebraic structure, for instance subgroups or subrings. This largely overlaps with our previous topic. Indeed a set in <code class="docutils literal notranslate">X</code> is implemented as a function from
- @@ -850,7 +852,7 @@ We won’t reuse <code class="docutils literal notranslate">X → Prop</code>. Instead there is a <code class="docutils literal notranslate">SetLike</code> class. Instead of wrapping an injection into a function type, that class wraps an injection into a <code class="docutils literal notranslate">Set</code> type and defines the corresponding coercion and <code class="docutils literal notranslate">Membership</code> instance. - <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] + <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] structure Submonoid₁ (M : Type) [Monoid M] where /-- The carrier of a submonoid. -/ carrier : Set M
html/C08_Groups_and_Rings.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>8. Groups and Rings — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -104,7 +106,7 @@
 <div itemprop="articleBody"> <section id="groups-and-rings"> - <h1>8. Groups and Rings<a class="headerlink" href="#groups-and-rings" title="Permalink to this heading"></a></h1> + <h1>8. Groups and Rings<a class="headerlink" href="#groups-and-rings" title="Link to this heading"></a></h1> We saw in <a class="reference internal" href="C02_Basics.html#proving-identities-in-algebraic-structures">Section 2.2</a> how to reason about operations in groups and rings. Later, in <a class="reference internal" href="C06_Structures.html#section-algebraic-structures">Section 6.2</a>, we saw how to define abstract algebraic structures, such as group structures, as well as concrete instances
- @@ -119,9 +121,9 @@ decisions behind the way the topics are treated.
 So making sense of some of the examples may require reviewing the background from <a class="reference internal" href="C07_Hierarchies.html#hierarchies">Chapter 7</a>. <section id="monoids-and-groups"> - <h2>8.1. Monoids and Groups<a class="headerlink" href="#monoids-and-groups" title="Permalink to this heading"></a></h2> - <section id="monoids-and-their-morphisms"> - <h3>8.1.1. Monoids and their morphisms<a class="headerlink" href="#monoids-and-their-morphisms" title="Permalink to this heading"></a></h3> + <h2>8.1. Monoids and Groups<a class="headerlink" href="#monoids-and-groups" title="Link to this heading"></a></h2> + <section id="monoids-and-their-morphisms"> + <h3>8.1.1. Monoids and their morphisms<a class="headerlink" href="#monoids-and-their-morphisms" title="Link to this heading"></a></h3> Courses in abstract algebra often start with groups and then progress to rings, fields, and vector spaces. This involves some contortions when discussing multiplication on rings since the multiplication operation does not come from a group structure
- @@ -171,7 +173,7 @@ composition to compose maps. Instead, we need to use <code class="docutils liter
 </div> </section> <section id="groups-and-their-morphisms"> - <h3>8.1.2. Groups and their morphisms<a class="headerlink" href="#groups-and-their-morphisms" title="Permalink to this heading"></a></h3> + <h3>8.1.2. Groups and their morphisms<a class="headerlink" href="#groups-and-their-morphisms" title="Link to this heading"></a></h3> We will have much more to say about groups, which are monoids with the extra property that every element has an inverse. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (x : G) : x * x⁻¹ = 1 := mul_inv_self x
- @@ -234,7 +236,7 @@ Doing so makes the inverse function noncomputable.
 </div> </section> <section id="subgroups"> - <h3>8.1.3. Subgroups<a class="headerlink" href="#subgroups" title="Permalink to this heading"></a></h3> + <h3>8.1.3. Subgroups<a class="headerlink" href="#subgroups" title="Link to this heading"></a></h3> Just as group morphisms are bundled, a subgroup of <code class="docutils literal notranslate">G</code> is also a bundled structure consisting of a set in <code class="docutils literal notranslate">G</code> with the relevant closure properties. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {G : Type*} [Group G] (H : Subgroup G) {x y : G} (hx : x ∈ H) (hy : y ∈ H) :
- @@ -400,29 +402,29 @@ Lean to use nonconstructive logic anyway using the following <code class="docuti
 open Subgroup - example {G : Type*} [Group G] [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime] + example {G : Type*} [Group G] [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime] (hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n := Sylow.exists_subgroup_card_pow_prime p hdvd </pre></div> </div> The next two exercises derive a corollary of Lagrange’s lemma. (This is also already in Mathlib, so do not use <code class="docutils literal notranslate">exact?</code> too quickly.) - <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} [Fintype H] : + <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} : H = ⊥ ↔ Nat.card H = 1 := by suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by - simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists, -Nat.card_eq_fintype_card] + simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists] sorry #check card_dvd_of_le - lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) [Fintype H] [Fintype K] + lemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G) (h : (Nat.card H).Coprime (Nat.card K)) : H ⊓ K = ⊥ := by sorry </pre></div> </div> </section> <section id="concrete-groups"> - <h3>8.1.4. Concrete groups<a class="headerlink" href="#concrete-groups" title="Permalink to this heading"></a></h3> + <h3>8.1.4. Concrete groups<a class="headerlink" href="#concrete-groups" title="Link to this heading"></a></h3> One can also manipulate concrete groups in Mathlib, although this is typically more complicated than working with the abstract theory. For instance, given any type <code class="docutils literal notranslate">X</code>, the group of permutations of <code class="docutils literal notranslate">X</code> is <code class="docutils literal notranslate">Equiv.Perm X</code>.
- @@ -499,7 +501,7 @@ to <code class="docutils literal notranslate">PresentedGroup.t
 </div> </section> <section id="group-actions"> - <h3>8.1.5. Group actions<a class="headerlink" href="#group-actions" title="Permalink to this heading"></a></h3> + <h3>8.1.5. Group actions<a class="headerlink" href="#group-actions" title="Link to this heading"></a></h3> One important way that group theory interacts with the rest of mathematics is through the use of group actions. An action of a group <code class="docutils literal notranslate">G</code> on some type <code class="docutils literal notranslate">X</code> is nothing more than a morphism from <code class="docutils literal notranslate">G</code> to
- @@ -597,7 +599,7 @@ conjugation, using our definition of <code class="docutils literal notranslate">
 </div> </section> <section id="quotient-groups"> - <h3>8.1.6. Quotient groups<a class="headerlink" href="#quotient-groups" title="Permalink to this heading"></a></h3> + <h3>8.1.6. Quotient groups<a class="headerlink" href="#quotient-groups" title="Link to this heading"></a></h3> In the above discussion of subgroups acting on groups, we saw the quotient <code class="docutils literal notranslate">G ⧸ H</code> appear. In general this is only a type. It can be endowed with a group structure such that the quotient map is a group morphism if and only if <code class="docutils literal notranslate">H</code> is a normal subgroup (and this group structure is
- @@ -666,7 +668,7 @@ or disjoint.
 #check Subgroup.index_mul_card #check Nat.eq_of_mul_eq_mul_right - lemma aux_card_eq [Fintype G] (h' : Nat.card G = Nat.card H * Nat.card K) : + lemma aux_card_eq [Finite G] (h' : Nat.card G = Nat.card H * Nat.card K) : Nat.card (G ⧸ H) = Nat.card K := by sorry </pre></div>
- @@ -702,9 +704,9 @@ morphisms from <code class="docutils literal notranslate">G&#8
 </section> </section> <section id="rings"> - <h2>8.2. Rings<a class="headerlink" href="#rings" title="Permalink to this heading"></a></h2> + <h2>8.2. Rings<a class="headerlink" href="#rings" title="Link to this heading"></a></h2> <section id="rings-their-units-morphisms-and-subrings"> - <h3>8.2.1. Rings, their units, morphisms and subrings<a class="headerlink" href="#rings-their-units-morphisms-and-subrings" title="Permalink to this heading"></a></h3> + <h3>8.2.1. Rings, their units, morphisms and subrings<a class="headerlink" href="#rings-their-units-morphisms-and-subrings" title="Link to this heading"></a></h3> The type of ring structures on a type <code class="docutils literal notranslate">R</code> is <code class="docutils literal notranslate">Ring R</code>. The variant where multiplication is assumed to be commutative is <code class="docutils literal notranslate">CommRing R</code>. We have already seen that the <code class="docutils literal notranslate">ring</code> tactic will prove any equality that follows from the axioms of a commutative ring.
- @@ -761,7 +763,7 @@ a subring.
 Also notice that <code class="docutils literal notranslate">RingHom.range</code> produces a subring. </section> <section id="ideals-and-quotients"> - <h3>8.2.2. Ideals and quotients<a class="headerlink" href="#ideals-and-quotients" title="Permalink to this heading"></a></h3> + <h3>8.2.2. Ideals and quotients<a class="headerlink" href="#ideals-and-quotients" title="Link to this heading"></a></h3> For historical reasons, Mathlib only has a theory of ideals for commutative rings. (The ring library was originally developed to make quick progress toward the foundations of modern algebraic geometry.) So in this section we will work with commutative (semi)rings.
- @@ -934,7 +936,7 @@ Remember that the <code class="docutils literal notranslate">r
 </div> </section> <section id="algebras-and-polynomials"> - <h3>8.2.3. Algebras and polynomials<a class="headerlink" href="#algebras-and-polynomials" title="Permalink to this heading"></a></h3> + <h3>8.2.3. Algebras and polynomials<a class="headerlink" href="#algebras-and-polynomials" title="Link to this heading"></a></h3> Given a commutative (semi)ring <code class="docutils literal notranslate">R</code>, an algebra over <code class="docutils literal notranslate">R</code> is a semiring <code class="docutils literal notranslate">A</code> equipped with a ring morphism whose image commutes with every element of <code class="docutils literal notranslate">A</code>. This is encoded as a type class <code class="docutils literal notranslate">Algebra R A</code>.
html/C09_Topology.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>9. Topology — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
- @@ -103,8 +105,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>9. Topology<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + <section id="index-0"> + <h1>9. Topology<a class="headerlink" href="#index-0" title="Link to this heading"></a></h1> Calculus is based on the concept of a function, which is used to model quantities that depend on one another. For example, it is common to study quantities that change over time.
- @@ -172,7 +174,7 @@ Formalizing mathematics requires making the relevant notion of “sameness&#
 fully explicit, and that is exactly what Bourbaki’s theory of filters manages to do. <section id="filters"> - <h2>9.1. Filters<a class="headerlink" href="#filters" title="Permalink to this heading"></a></h2> + <h2>9.1. Filters<a class="headerlink" href="#filters" title="Link to this heading"></a></h2> A filter on a type <code class="docutils literal notranslate">X</code> is a collection of sets of <code class="docutils literal notranslate">X</code> that satisfies three conditions that we will spell out below. The notion supports two related ideas:
- @@ -526,7 +528,7 @@ by definition, the assumption <code class="docutils literal notranslate"> </section> <section id="metric-spaces"> - <h2>9.2. Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this heading"></a></h2> + <h2>9.2. Metric spaces<a class="headerlink" href="#metric-spaces" title="Link to this heading"></a></h2> Examples in the previous section focus on sequences of real numbers. In this section we will go up a bit in generality and focus on metric spaces. A metric space is a type <code class="docutils literal notranslate">X</code> equipped with a distance function <code class="docutils literal notranslate">dist : X → X → ℝ</code> which is a generalization of the function <code class="docutils literal notranslate">fun x y ↦ |x - y|</code> from the case where <code class="docutils literal notranslate">X = ℝ</code>.
- @@ -545,7 +547,7 @@ They are called <code class="docutils literal notranslate">EMe
 Note that our journey from <code class="docutils literal notranslate">ℝ</code> to metric spaces jumped over the special case of normed spaces that also require linear algebra and will be explained as part of the calculus chapter. <section id="convergence-and-continuity"> - <h3>9.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this heading"></a></h3> + <h3>9.2.1. Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Link to this heading"></a></h3> Using distance functions, we can already define convergent sequences and continuous functions between metric spaces. They are actually defined in a more general setting covered in the next section, but we have lemmas recasting the definition in terms of distances.
- @@ -634,7 +636,7 @@ and get our final proof, now bordering obfuscation.
 </div> </section> <section id="balls-open-sets-and-closed-sets"> - <h3>9.2.2. Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this heading"></a></h3> + <h3>9.2.2. Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Link to this heading"></a></h3> Once we have a distance function, the most important geometric definitions are (open) balls and closed balls. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable (r : ℝ)
- @@ -691,7 +693,7 @@ argument so we can invoke <code class="docutils literal notranslate"> </section> <section id="compactness"> - <h3>9.2.3. Compactness<a class="headerlink" href="#compactness" title="Permalink to this heading"></a></h3> + <h3>9.2.3. Compactness<a class="headerlink" href="#compactness" title="Link to this heading"></a></h3> Compactness is an important topological notion. It distinguishes subsets of a metric space that enjoy the same kind of properties as segments in reals compared to other intervals: <ul class="simple">
- @@ -734,7 +736,7 @@ are deduced from more general versions, some of which will be discussed in later
 In a compact metric space any closed set is compact, this is <code class="docutils literal notranslate">IsClosed.isCompact</code>. </section> <section id="uniformly-continuous-functions"> - <h3>9.2.4. Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this heading"></a></h3> + <h3>9.2.4. Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Link to this heading"></a></h3> We now turn to uniformity notions on metric spaces : uniformly continuous functions, Cauchy sequences and completeness. Again those are defined in a more general context but we have lemmas in the metric name space to access their elementary definitions. We start with uniform continuity.
- @@ -765,7 +767,7 @@ of the distance function on <code class="docutils literal notranslate"> </section> <section id="completeness"> - <h3>9.2.5. Completeness<a class="headerlink" href="#completeness" title="Permalink to this heading"></a></h3> + <h3>9.2.5. Completeness<a class="headerlink" href="#completeness" title="Link to this heading"></a></h3> A Cauchy sequence in a metric space is a sequence whose terms get closer and closer to each other. There are a couple of equivalent ways to state that idea. In particular converging sequences are Cauchy. The converse is true only in so-called complete
- @@ -857,9 +859,9 @@ define something inductively in the middle of a proof using <code class="docutil
 </section> </section> <section id="topological-spaces"> - <h2>9.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> + <h2>9.3. Topological spaces<a class="headerlink" href="#topological-spaces" title="Link to this heading"></a></h2> <section id="fundamentals"> - <h3>9.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> + <h3>9.3.1. Fundamentals<a class="headerlink" href="#fundamentals" title="Link to this heading"></a></h3> We now go up in generality and introduce topological spaces. We will review the two main ways to define topological spaces and then explain how the category of topological spaces is much better behaved than the category of metric spaces. Note that we won’t be using Mathlib category theory here, only having
- @@ -1049,7 +1051,7 @@ Let us explore that constraint “on paper” using notation </section> <section id="separation-and-countability"> - <h3>9.3.2. Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this heading"></a></h3> + <h3>9.3.2. Separation and countability<a class="headerlink" href="#separation-and-countability" title="Link to this heading"></a></h3> We saw that the category of topological spaces have very nice properties. The price to pay for this is existence of rather pathological topological spaces. There are a number of assumptions you can make on a topological space to ensure its behavior
- @@ -1138,7 +1140,7 @@ of sets can be understood using sequences.
 </div> </section> <section id="id5"> - <h3>9.3.3. Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> + <h3>9.3.3. Compactness<a class="headerlink" href="#id5" title="Link to this heading"></a></h3> Let us now discuss how compactness is defined for topological spaces. As usual there are several ways to think about it and Mathlib goes for the filter version. We first need to define cluster points of filters. Given a filter <code class="docutils literal notranslate">F</code> on a topological space <code class="docutils literal notranslate">X</code>,
html/C10_Differential_Calculus.html
- @@ -1,23 +1,25 @@
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- @@ -96,8 +98,8 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>10. Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + <section id="index-0"> + <h1>10. Differential Calculus<a class="headerlink" href="#index-0" title="Link to this heading"></a></h1> We now consider the formalization of notions from analysis, starting with differentiation in this chapter and turning integration and measure theory in the next.
- @@ -107,7 +109,7 @@ which is familiar from any introductory calculus class.
 In <a class="reference internal" href="#normed-spaces">Section 10.2</a>, we then consider the notion of a derivative in a much broader setting. <section id="elementary-differential-calculus"> - <h2>10.1. Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this heading"></a></h2> + <h2>10.1. Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Link to this heading"></a></h2> Let <code class="docutils literal notranslate">f</code> be a function from the reals to the reals. There is a difference between talking about the derivative of <code class="docutils literal notranslate">f</code> at a single point and talking about the derivative function.
- @@ -182,12 +184,12 @@ seems even weirder.
 </div> </section> <section id="differential-calculus-in-normed-spaces"> - <h2>10.2. Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this heading"></a></h2> + <h2>10.2. Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Link to this heading"></a></h2> <section id="id3"> - <h3>10.2.1. Normed spaces<a class="headerlink" href="#id3" title="Permalink to this heading"></a></h3> + <h3>10.2.1. Normed spaces<a class="headerlink" href="#id3" title="Link to this heading"></a></h3> Differentiation can be generalized beyond <code class="docutils literal notranslate">ℝ</code> using the notion of a normed vector space, which encapsulates both direction and distance. - We start with the notion of a normed group, which as an additive commutative + We start with the notion of a normed group, which is an additive commutative group equipped with a real-valued norm function satisfying the following conditions. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {E : Type*} [NormedAddCommGroup E]
- @@ -249,7 +251,7 @@ complete as long as the field itself is complete.
 </div> </section> <section id="continuous-linear-maps"> - <h3>10.2.2. Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this heading"></a></h3> + <h3>10.2.2. Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Link to this heading"></a></h3> We now turn to the morphisms in the category of normed spaces, namely, continuous linear maps. In Mathlib, the type of <code class="docutils literal notranslate">𝕜</code>-linear continuous maps between normed spaces
- @@ -299,7 +301,7 @@ bounded, then the norms of these linear maps are uniformly bounded.
 The main ingredient is Baire’s theorem <code class="docutils literal notranslate">nonempty_interior_of_iUnion_of_closed</code>. (You proved a version of this in the topology chapter.) Minor ingredients include <code class="docutils literal notranslate">continuous_linear_map.opNorm_le_of_shell</code>, - <code class="docutils literal notranslate">interior_subset</code> and <code class="docutils literal notranslate">interior_iInter_subset</code> and <code class="docutils literal notranslate">is_closed_le</code>. + <code class="docutils literal notranslate">interior_subset</code> and <code class="docutils literal notranslate">interior_iInter_subset</code> and <code class="docutils literal notranslate">isClosed_le</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
- @@ -331,7 +333,7 @@ Minor ingredients include <code class="docutils literal notranslate"> </section> <section id="asymptotic-comparisons"> - <h3>10.2.3. Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this heading"></a></h3> + <h3>10.2.3. Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Link to this heading"></a></h3> Defining differentiability also requires asymptotic comparisons. Mathlib has an extensive library covering the big O and little o relations, whose definitions are shown below.
- @@ -359,7 +361,7 @@ Here we will only use little o to define differentiability.
 </div> </section> <section id="differentiability"> - <h3>10.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> + <h3>10.2.4. Differentiability<a class="headerlink" href="#differentiability" title="Link to this heading"></a></h3> We are now ready to discuss differentiable functions between normed spaces. In analogy the elementary one-dimensional, Mathlib defines a predicate <code class="docutils literal notranslate">HasFDerivAt</code> and a function <code class="docutils literal notranslate">fderiv</code>.
html/C11_Integration_and_Measure_Theory.html
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- @@ -91,10 +93,10 @@
 <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="index-0"> - <h1>11. Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + <section id="index-0"> + <h1>11. Integration and Measure Theory<a class="headerlink" href="#index-0" title="Link to this heading"></a></h1> <section id="elementary-integration"> - <h2>11.1. Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this heading"></a></h2> + <h2>11.1. Elementary Integration<a class="headerlink" href="#elementary-integration" title="Link to this heading"></a></h2> We first focus on integration of functions on finite intervals in <code class="docutils literal notranslate">ℝ</code>. We can integrate elementary functions. <div class="highlight-lean notranslate"><div class="highlight"><pre>open MeasureTheory intervalIntegral
- @@ -133,7 +135,7 @@ which are not shown here, are not equivalent.)
 </div> </section> <section id="measure-theory"> - <h2>11.2. Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this heading"></a></h2> + <h2>11.2. Measure Theory<a class="headerlink" href="#measure-theory" title="Link to this heading"></a></h2> The general context for integration in Mathlib is measure theory. Even the elementary integrals of the previous section are in fact Bochner integrals. Bochner integration is a generalization of Lebesgue integration where the target space can be any Banach space,
- @@ -210,7 +212,7 @@ almost everywhere.
 </div> </section> <section id="integration"> - <h2>11.3. Integration<a class="headerlink" href="#integration" title="Permalink to this heading"></a></h2> + <h2>11.3. Integration<a class="headerlink" href="#integration" title="Link to this heading"></a></h2> Now that we have measurable spaces and measures we can consider integrals. As explained above, Mathlib uses a very general notion of integration that allows any Banach space as the target.
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html/_static/documentation_options.js
- @@ -1,5 +1,4 @@
  - var DOCUMENTATION_OPTIONS = { - URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'), + const DOCUMENTATION_OPTIONS = { VERSION: '0.1', LANGUAGE: 'en', COLLAPSE_INDEX: false,
html/_static/pygments.css
- @@ -17,6 +17,7 @@ span.linenos.special { color: #000000; background-color: #ffffc0; padding-left:
  .highlight .cs { color: #3D7B7B; font-style: italic } /* Comment.Special */ .highlight .gd { color: #A00000 } /* Generic.Deleted */ .highlight .ge { font-style: italic } /* Generic.Emph */ + .highlight .ges { font-weight: bold; font-style: italic } /* Generic.EmphStrong */ .highlight .gr { color: #E40000 } /* Generic.Error */ .highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */ .highlight .gi { color: #008400 } /* Generic.Inserted */
html/_static/searchtools.js
- @@ -57,12 +57,12 @@ const _removeChildren = (element) => {
  const _escapeRegExp = (string) => string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string - const _displayItem = (item, searchTerms) => { + const _displayItem = (item, searchTerms, highlightTerms) => { const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; - const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; + const contentRoot = document.documentElement.dataset.content_root; const [docName, title, anchor, descr, score, _filename] = item;
- @@ -75,20 +75,24 @@ const _displayItem = (item, searchTerms) => {
  if (dirname.match(/\/index\/$/)) dirname = dirname.substring(0, dirname.length - 6); else if (dirname === "index/") dirname = ""; - requestUrl = docUrlRoot + dirname; + requestUrl = contentRoot + dirname; linkUrl = requestUrl; } else { // normal html builders - requestUrl = docUrlRoot + docName + docFileSuffix; + requestUrl = contentRoot + docName + docFileSuffix; linkUrl = docName + docLinkSuffix; } let linkEl = listItem.appendChild(document.createElement("a")); linkEl.href = linkUrl + anchor; linkEl.dataset.score = score; linkEl.innerHTML = title; - if (descr) + if (descr) { listItem.appendChild(document.createElement("span")).innerHTML = " (" + descr + ")"; + // highlight search terms in the description + if (SPHINX_HIGHLIGHT_ENABLED) // set in sphinx_highlight.js + highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted")); + } else if (showSearchSummary) fetch(requestUrl) .then((responseData) => responseData.text())
- @@ -97,6 +101,9 @@ const _displayItem = (item, searchTerms) => {
  listItem.appendChild( Search.makeSearchSummary(data, searchTerms) ); + // highlight search terms in the summary + if (SPHINX_HIGHLIGHT_ENABLED) // set in sphinx_highlight.js + highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted")); }); Search.output.appendChild(listItem); };
- @@ -115,14 +122,15 @@ const _finishSearch = (resultCount) => {
  const _displayNextItem = ( results, resultCount, - searchTerms + searchTerms, + highlightTerms, ) => { // results left, load the summary and display it // this is intended to be dynamic (don't sub resultsCount) if (results.length) { - _displayItem(results.pop(), searchTerms); + _displayItem(results.pop(), searchTerms, highlightTerms); setTimeout( - () => _displayNextItem(results, resultCount, searchTerms), + () => _displayNextItem(results, resultCount, searchTerms, highlightTerms), 5 ); }
- @@ -360,7 +368,7 @@ const Search = {
  // console.info("search results:", Search.lastresults); // print the results - _displayNextItem(results, results.length, searchTerms); + _displayNextItem(results, results.length, searchTerms, highlightTerms); }, /**
html/_static/sphinx_highlight.js
- @@ -29,14 +29,19 @@ const _highlight = (node, addItems, text, className) => {
  } span.appendChild(document.createTextNode(val.substr(pos, text.length))); + const rest = document.createTextNode(val.substr(pos + text.length)); parent.insertBefore( span, parent.insertBefore( - document.createTextNode(val.substr(pos + text.length)), + rest, node.nextSibling ) ); node.nodeValue = val.substr(0, pos); + /* There may be more occurrences of search term in this node. So call this + * function recursively on the remaining fragment. + */ + _highlight(rest, addItems, text, className); if (isInSVG) { const rect = document.createElementNS(
- @@ -140,5 +145,10 @@ const SphinxHighlight = {
  }, }; - _ready(SphinxHighlight.highlightSearchWords); - _ready(SphinxHighlight.initEscapeListener); + _ready(() => { + /* Do not call highlightSearchWords() when we are on the search page. + * It will highlight words from the *previous* search query. + */ + if (typeof Search === "undefined") SphinxHighlight.highlightSearchWords(); + SphinxHighlight.initEscapeListener(); + });
html/genindex.html
- @@ -1,22 +1,24 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>Index — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="#" /> <link rel="search" title="Search" href="search.html" />
html/index.html
- @@ -1,23 +1,25 @@
 <!DOCTYPE html> - <html class="writer-html5" lang="en" > + <html class="writer-html5" lang="en" data-content_root="./"> <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>Mathematics in Lean — Mathematics in Lean 0.1 documentation</title> - <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> - <link rel="stylesheet" href="_static/css/custom.css" type="text/css" /> + <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" /> + <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" /> + <link rel="stylesheet" type="text/css" href="_static/css/custom.css?v=0731ccc3" /> + + <link rel="shortcut icon" href="_static/favicon.ico"/>  - <script src="_static/jquery.js"></script> - <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> - <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> - <script src="_static/doctools.js"></script> - <script src="_static/sphinx_highlight.js"></script> + <script src="_static/jquery.js?v=5d32c60e"></script> + <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script> + <script src="_static/documentation_options.js?v=2709fde1"></script> + <script src="_static/doctools.js?v=888ff710"></script> + <script src="_static/sphinx_highlight.js?v=dc90522c"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" />
- @@ -85,7 +87,7 @@
 <div itemprop="articleBody"> <section id="mathematics-in-lean"> - <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this heading"></a></h1> + <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Link to this heading"></a></h1> <div class="toctree-wrapper compound"> <ul> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a><ul>
html/objects.inv
html/search.html
- @@ -1,23 +1,25 @@
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html/searchindex.js
- @@ -1,1 +1,1 @@
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lake-manifest.json
- @@ -5,7 +5,7 @@
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- @@ -35,27 +35,27 @@
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lean-toolchain
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  - leanprover/lean4:v4.10.0-rc2 + leanprover/lean4:v4.10.0
mathematics_in_lean.pdf